3.73 \(\int \frac{x}{\sqrt{-1+x^3} \left (8+x^3\right )} \, dx\)

Optimal. Leaf size=74 \[ \frac{1}{18} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right )+\frac{1}{18} \tan ^{-1}\left (\frac{\sqrt{x^3-1}}{3}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (1-x)}{\sqrt{x^3-1}}\right )}{6 \sqrt{3}} \]

[Out]

ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])]/18 + ArcTan[Sqrt[-1 + x^3]/3]/18 - ArcTanh[
(Sqrt[3]*(1 - x))/Sqrt[-1 + x^3]]/(6*Sqrt[3])

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Rubi [A]  time = 0.288252, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{1}{18} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right )+\frac{1}{18} \tan ^{-1}\left (\frac{\sqrt{x^3-1}}{3}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (1-x)}{\sqrt{x^3-1}}\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(Sqrt[-1 + x^3]*(8 + x^3)),x]

[Out]

ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])]/18 + ArcTan[Sqrt[-1 + x^3]/3]/18 - ArcTanh[
(Sqrt[3]*(1 - x))/Sqrt[-1 + x^3]]/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 4.32494, size = 37, normalized size = 0.5 \[ - \frac{x^{2} \sqrt{x^{3} - 1} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},x^{3},- \frac{x^{3}}{8} \right )}}{16 \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**3+8)/(x**3-1)**(1/2),x)

[Out]

-x**2*sqrt(x**3 - 1)*appellf1(2/3, 1/2, 1, 5/3, x**3, -x**3/8)/(16*sqrt(-x**3 +
1))

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Mathematica [C]  time = 0.191349, size = 118, normalized size = 1.59 \[ -\frac{20 x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};x^3,-\frac{x^3}{8}\right )}{\sqrt{x^3-1} \left (x^3+8\right ) \left (3 x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};x^3,-\frac{x^3}{8}\right )-4 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};x^3,-\frac{x^3}{8}\right )\right )-40 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};x^3,-\frac{x^3}{8}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/(Sqrt[-1 + x^3]*(8 + x^3)),x]

[Out]

(-20*x^2*AppellF1[2/3, 1/2, 1, 5/3, x^3, -x^3/8])/(Sqrt[-1 + x^3]*(8 + x^3)*(-40
*AppellF1[2/3, 1/2, 1, 5/3, x^3, -x^3/8] + 3*x^3*(AppellF1[5/3, 1/2, 2, 8/3, x^3
, -x^3/8] - 4*AppellF1[5/3, 3/2, 1, 8/3, x^3, -x^3/8])))

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Maple [C]  time = 0.169, size = 286, normalized size = 3.9 \[ -{\frac{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}{9}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{i}{6}}\sqrt{3}+{\frac{1}{2}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}}+{\frac{\sqrt{2}}{36}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{2}-2\,{\it \_Z}+4 \right ) }{ \left ( 2-{\it \_alpha} \right ) \left ( -1+{\it \_alpha} \right ) \left ( -i\sqrt{3}-3 \right ) \sqrt{{\frac{-1+x}{-i\sqrt{3}-3}}}\sqrt{{\frac{2\,x+1-i\sqrt{3}}{-i\sqrt{3}+3}}}\sqrt{{\frac{2\,x+1+i\sqrt{3}}{i\sqrt{3}+3}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{i}{6}}{\it \_alpha}\,\sqrt{3}+{\frac{{\it \_alpha}}{2}}-{\frac{i}{6}}\sqrt{3}-{\frac{1}{2}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^3+8)/(x^3-1)^(1/2),x)

[Out]

-1/9*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1
/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2
)/(x^3-1)^(1/2)*EllipticPi(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/6*I*3^(1/2)+1/2
,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+1/36*2^(1/2)*sum((2-_alpha)*(-
1+_alpha)*(-I*3^(1/2)-3)*((-1+x)/(-I*3^(1/2)-3))^(1/2)*((2*x+1-I*3^(1/2))/(-I*3^
(1/2)+3))^(1/2)*((2*x+1+I*3^(1/2))/(I*3^(1/2)+3))^(1/2)/(x^3-1)^(1/2)*EllipticPi
(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/6*I*_alpha*3^(1/2)+1/2*_alpha-1/6*I*3^(1/
2)-1/2,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)),_alpha=RootOf(_Z^2-2*_Z+
4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{3} + 8\right )} \sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^3 + 8)*sqrt(x^3 - 1)),x, algorithm="maxima")

[Out]

integrate(x/((x^3 + 8)*sqrt(x^3 - 1)), x)

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Fricas [A]  time = 0.342107, size = 950, normalized size = 12.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^3 + 8)*sqrt(x^3 - 1)),x, algorithm="fricas")

[Out]

1/216*sqrt(3)*log(3*(x^6 + 48*x^5 + 186*x^4 - 56*x^3 + 6*sqrt(3)*(x^4 + 12*x^3 +
12*x^2 - 16*x)*sqrt(x^3 - 1) - 120*x^2 - 96*x + 64)/(x^6 - 6*x^5 + 24*x^4 - 56*
x^3 + 96*x^2 - 96*x + 64)) - 1/216*sqrt(3)*log(3*(x^6 + 48*x^5 + 186*x^4 - 56*x^
3 - 6*sqrt(3)*(x^4 + 12*x^3 + 12*x^2 - 16*x)*sqrt(x^3 - 1) - 120*x^2 - 96*x + 64
)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)) - 1/54*arctan(3*(9*x^5 -
9*x^4 - 54*x^3 + sqrt(3)*(x^4 + 8*x^3 - 30*x^2 - 4*x + 16)*sqrt(x^3 - 1) + 18*x
^2 + 36*x)/(sqrt(3)*(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)*sqrt((x
^6 + 48*x^5 + 186*x^4 - 56*x^3 + 6*sqrt(3)*(x^4 + 12*x^3 + 12*x^2 - 16*x)*sqrt(x
^3 - 1) - 120*x^2 - 96*x + 64)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x +
64)) + sqrt(3)*(x^6 + 3*x^5 - 75*x^4 + 88*x^3 + 6*x^2 + 84*x - 80) + 9*(x^4 - 6*
x^3 - 6*x^2 + 20*x)*sqrt(x^3 - 1))) - 1/54*arctan(-3*(9*x^5 - 9*x^4 - 54*x^3 - s
qrt(3)*(x^4 + 8*x^3 - 30*x^2 - 4*x + 16)*sqrt(x^3 - 1) + 18*x^2 + 36*x)/(sqrt(3)
*(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)*sqrt((x^6 + 48*x^5 + 186*x
^4 - 56*x^3 - 6*sqrt(3)*(x^4 + 12*x^3 + 12*x^2 - 16*x)*sqrt(x^3 - 1) - 120*x^2 -
96*x + 64)/(x^6 - 6*x^5 + 24*x^4 - 56*x^3 + 96*x^2 - 96*x + 64)) + sqrt(3)*(x^6
+ 3*x^5 - 75*x^4 + 88*x^3 + 6*x^2 + 84*x - 80) - 9*(x^4 - 6*x^3 - 6*x^2 + 20*x)
*sqrt(x^3 - 1))) + 1/54*arctan(1/6*(x^3 - 12*x^2 - 6*x - 10)/(sqrt(x^3 - 1)*(x -
1)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 2\right ) \left (x^{2} - 2 x + 4\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**3+8)/(x**3-1)**(1/2),x)

[Out]

Integral(x/(sqrt((x - 1)*(x**2 + x + 1))*(x + 2)*(x**2 - 2*x + 4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{3} + 8\right )} \sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((x^3 + 8)*sqrt(x^3 - 1)),x, algorithm="giac")

[Out]

integrate(x/((x^3 + 8)*sqrt(x^3 - 1)), x)